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Properties

Label 1-4851-4851.1303-r0-0-0
Degree $1$
Conductor $4851$
Sign $-0.805 + 0.593i$
Analytic cond. $22.5279$
Root an. cond. $22.5279$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.251 + 0.967i)2-s + (−0.873 + 0.486i)4-s + (−0.646 + 0.762i)5-s + (−0.691 − 0.722i)8-s + (−0.900 − 0.433i)10-s + (0.712 − 0.701i)13-s + (0.525 − 0.850i)16-s + (−0.393 + 0.919i)17-s + (0.309 + 0.951i)19-s + (0.193 − 0.981i)20-s + (−0.733 − 0.680i)23-s + (−0.163 − 0.986i)25-s + (0.858 + 0.512i)26-s + (0.420 + 0.907i)29-s + (−0.104 − 0.994i)31-s + (0.955 + 0.294i)32-s + ⋯
L(s)  = 1  + (0.251 + 0.967i)2-s + (−0.873 + 0.486i)4-s + (−0.646 + 0.762i)5-s + (−0.691 − 0.722i)8-s + (−0.900 − 0.433i)10-s + (0.712 − 0.701i)13-s + (0.525 − 0.850i)16-s + (−0.393 + 0.919i)17-s + (0.309 + 0.951i)19-s + (0.193 − 0.981i)20-s + (−0.733 − 0.680i)23-s + (−0.163 − 0.986i)25-s + (0.858 + 0.512i)26-s + (0.420 + 0.907i)29-s + (−0.104 − 0.994i)31-s + (0.955 + 0.294i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.805 + 0.593i$
Analytic conductor: \(22.5279\)
Root analytic conductor: \(22.5279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (1303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (0:\ ),\ -0.805 + 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4153890532 + 1.264270567i\)
\(L(\frac12)\) \(\approx\) \(0.4153890532 + 1.264270567i\)
\(L(1)\) \(\approx\) \(0.7460029033 + 0.6087507376i\)
\(L(1)\) \(\approx\) \(0.7460029033 + 0.6087507376i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.251 + 0.967i)T \)
5 \( 1 + (-0.646 + 0.762i)T \)
13 \( 1 + (0.712 - 0.701i)T \)
17 \( 1 + (-0.393 + 0.919i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (0.420 + 0.907i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.995 - 0.0896i)T \)
41 \( 1 + (0.971 - 0.237i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
47 \( 1 + (0.998 + 0.0598i)T \)
53 \( 1 + (-0.393 - 0.919i)T \)
59 \( 1 + (-0.280 - 0.959i)T \)
61 \( 1 + (0.992 - 0.119i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.753 - 0.657i)T \)
73 \( 1 + (-0.550 - 0.834i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.712 + 0.701i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.83774410213074663895861095732, −17.448880729800655955914835676011, −16.41598373892036861614211297776, −15.7286776953743719709857050873, −15.34481496651205603387623282287, −14.13066066178761957029983812343, −13.71111235839520406120510891414, −13.18368049901913807926135726333, −12.17160644688325083387998893531, −11.92126369450703729905227417110, −11.21526159301969759080007451862, −10.61276199790031161408369378714, −9.59693978772027842538301384712, −9.06004736806949463993827400513, −8.58886062711181588551636087195, −7.66950895574147491917062289441, −6.83670721672354240121498442117, −5.80385000163019448943906845307, −5.09971460182817722276937455123, −4.400723954726840929151776705156, −3.873248381166499074160151080167, −3.03431626928152849977126784211, −2.15589799561205373677815653267, −1.272197825293467407402981271806, −0.48900195074036469083641944908, 0.7539187763883967945158477938, 2.10776843103947065286020249722, 3.20785735888517860126759625863, 3.73985757474937773844586116284, 4.34092818638647895337065800470, 5.35287664670270064577035870251, 6.13828223739804070598114806927, 6.521001486349202416355128155870, 7.456623415158378853315182521238, 8.05506599666279109106859800331, 8.46107048155601427025301300306, 9.42368612108189152304504878208, 10.356201818350913746876940205357, 10.79788166298037917714421415108, 11.7941872401575315865075654468, 12.4940366000354598122975345638, 13.04243438927158300863129755141, 13.964189838646459386170249362074, 14.494474428182478278485744607716, 15.00747965595920811681699184873, 15.82860650591039232778775120993, 16.074143601323172842102958740809, 16.95304062994500411212390122974, 17.80807023846102771724235758888, 18.16108752163681089826587523900

Graph of the $Z$-function along the critical line